
where we have used \( d{\bf x}_t - A({\bf x}_t) {\text{d}}t = {\bf u}({\bf x}_t,t) {\text{d}}t+ {\text{d}}{\bf W}_t\) , with \({\text{d}}{\bf W}_t\)  as the stochastic differential measure for a multi-variate Wiener process (see ref. [1]}). Importantly, with the criteria of path integral control (i.e., \(K^{-1} = B/\lambda \) ), the Kullback-Leibler distance between the artificially and naturally selected phenotype distributions is equivalent to the cumulative cost of intervention, divided by the overall cost of artificial selection \(\lambda \) ,
\(D_{KL} (P_{\bf u}({\bf x}_{t_0}^{t_f}) || P_0({\bf x}_{t_0}^{t_f})) = \frac{1}{2\lambda } \left\langle \left({\bf u}_{t_0}^{t_f}\right)^\top B\, {\bf u}_{t_0}^{t_f}\right\rangle \) 
